



Class Q/- __., 



Book 



-> — «D ^ 






COPYRIGHT DEPOSIT. 



THE 

INTEGKAL CALCULUS 

ON THE INTEGRATION OF 

THE POWERS OF TRANSCENDENTAL FUNC- 
TIONS, NEW METHODS AND THEOREMS, 
CALCULATION OF THE BERNOULLIAN 
NUMBERS, RECTIFICATION OF THE 
LOGARITHMIC CURVE, INTE- 
GRATION OF LOGARITHMIC 
BINOMIALS, ETC. 



BY 
JAMES BALLANTYNE 



PUBLISHED BY THE AUTHOR 

6 GREENVILLE ST., BOSTON 

1919 






\ 



COPYRIGHTED, 1919, 
BY JAMES BALLANTYNE 



NOV -8 1320 



THEPLIMPTON-PRESS 
NOaWOOD-MASS-U-S-A 

©CU601482 



CONTENTS 

PAGE 

Section One: 

Tables and Series for the Integration of the Powers of Cir- 
cular Functions 5 

Section Two: 

Some Well-known Formulae Derived from these Integrals. 11 

Section Three: 

Series for Expressing the Tangent in Terms of the Arc, 

Derived from Different Sources 16 

The Law of Coefficients, and Its Relation to the Bernoul- 

lian Numbers 17 

Method of Calculating the Bernoullian Numbers 19 

Table of these Numbers and Numerators of Coefficients. 22 

Section Four: 

Theorem for Integration of the Powers of Transcendental 

Functions in Terms of the Functions 23 

Logarithmic Integrals, for any System of Logarithms ... 26 
Rectification of the Logarithmic Curve, etc 29 

Section Five: 

Integration of Logarithmic Binomials, etc 34 

Theorem for Integrating certain Classes of Algebraic Frac- 
tions, and Fractions Containing Circular Functions ... 37 



THE INTEGRAL CALCULUS 



SECTION ONE 

On the Integration of the Powers of Trigonometrical Functions 

1. In performing the operation of integrating a differential it 
is usual to add the constant C to the result, because the integral 
from which the differential has been derived may contain some 
constant quantity not affected by the variable; in which case the 
constant would not be indicated in the differential. In the follow- 
ing Tables and Series the constant has been fully accounted for, 
and the complete integrals are therein expressed, excepting in those 
cases where the constant is particularly noted. 



Table I 
Integrals of Even Powers of sin x • dx y to Radius 1 

fsmPx-dx = x. 

x — cos x • sin x 



J*sm 2 X'dx = 
fsmfy'dx = 
J*sin G X'dx = 



2 

3x — 3 cos x - sin x — 2 cos x • sin 3 :r 
2^4 

15 x — 15 cos z-sin x — 10 cos z-sin 3 £ — 8 cos x- sin 5 x 

2-4-6 



fsufix-dx = (105 x — 105 cos x • sin x — 70 cos #-sin 3 £ 

— 56 cos x • sin 5 x — 48 cos x • sin 7 x) ~ 2 • 4 • 6 • 8 

y*sin 10 X'dx = (945 x — 945 cos z-sin # — 630 cos #-sin 3 £ 

— 504 cos x • sin 5 # — 432 cos x ■ sm 7 x — 384 cos x • sin 9 #) 
-2-4- 6-8-10 

2. This table may be carried to any extent by observing the fol- 
lowing law of progression : 

n • i 171 — 1 C • 07 1 ■ m i 

f%m m x • dx = I sin m-2 # -dx cos x • sin m_1 ^, 

m J m 



6 TH"F INTEGRAL CALCULUS 

and the table may be extended to negative values of m, by changing 
this expression to 

rsm m ~ 2 X'dx = 7 | sm m x-dx ^ cos x • sin" 1-1 ^ ; thus, 

m - 1 J m - 1 

f sin -2 £ • dx = — cot x + C. 

cot a; 

2 COt x + shr^ 
f sin~ 4 :r • dx = = + C. 

„ 4 cot x 3 cot x 

8 cot x H . _ H —. — 

n . , 7 sm 2 x sm 4 z ~ 

y sui~ 6 x • dx = — h c . 

3-o 

24 cot z 18 cot x 15 cot x 

48 cot £ H ^ 1 =-7 h — —- — 

n . a 7 sin 2 x sin 4 x sin b z ~ 
fsw.-*x>dx = 3.5.7 *" 

The value of C is the area of the full quadrant of the curvilinear 
for each particular value of m; that is, 

IT 

C = I sm m x-dx. 

Table II 
Integrals of Even Powers of cos x • dx, to Radius 1 

J*cos°x-dx = x. 

n «, x + sin x • cos x 
J cos 2 z • dx = — 



fcos A X'dx = 
J*cos 6 x-dx = 



2 

3x + 3 sin x • cos z + 2 sin z • cos 3 z 
2^4 

15x + 15 sin x • cos z 4- 10 sin x • cos 3 x + 8 sin x • cos 5 x 

2-4-6 



fco&X'dx = (105z + 105 sin z-cos x + 70 sin z-cos 3 :r 

+ 56 sin £ • cos 5 z + 48 sin x • cos 7 x) -s- 2-4-6-8. 
y'cos 10 ^ • c/x = (945 x + 945 sin z • cos x + 630 sin x • cos 3 z -+- 

504 sin x • cos 5 x + 432 sin x • cos 7 z + 384 sin x • cos 9 z) 
- 2-4-6-8-10 
3. This table may be extended by the following law of progression: 

m — 1 C 1 • 

rcos m x-dx = / cos m ~ 2 x-dx + — sin x • cos 771-1 ^, 

raj m 

and for negative values of m, this expression may be changed to 

m C 1 

rcos m ~ 2 X'dx = 7 I cos m x-dx ; sin x • cos m-1 a;, 

m — 1 J m — 1 



THE INTEGRAL CALCULUS 

and from this we get 

J^cos^x-^x = tan x. 

n . tan x 

y 2 tan x + - — =- 

r -a j cos ^ 

4 tan x 3 tan re 

8 tan x -\ -„ 1 — 

_ . . cos 2 rc cos 4 rc 

jcos~ 6 X'dx = 



3-5 

. n , 24 tan x 18 tan a: 15 tan re 

48 tan x + = — + — + — 

„ a 7 cos 2 re cos 4 rc cos 6 rc 
fcosr*x • ax = 3.5.7 

Table III 
Integrals of Odd Negative Powers of Cos x • dx y to Radius 1 

1 + sin x 



y* cos -1 z • dx = I log 



1 - sin x 



_ , , , 1 + sin x tan x 

JCOS~ 3 X-dX = § log : + 



3 



1 — sin x cos x 

, 1 + sin x 3 tan x 2 tan x 
lo S 1 ~zrz + — — - + 



„ K , 2 1 — sin x cos x cos 3 rc 
jcos~ 5 x-dx = 2T4 

, _ , 1 + sin x 15 tan re 10 tan x 8 tan x 

¥■ log 1 = — + + — + — r" 

„ , , 1 - sin a; cos re cos^rc cos 5 re 
fcos- 7 x-dx = 2^6 

The law of progression is 

/l , m C 1 7 1 tan x 
-TTT'dx = 7 I ax H 7 
cos w+2 rc m - 1 J cos m re m + 1 cos m re 

Table IV 
Integrals of Odd Negative Powers of Sin x • dx, to Radius 1. 

r$\rr l X'dx = -§ log - + C. 

' 1- COS X 

, , 1 + COS x cot x 

h log - h 

„ . , 7 1- cos # sin 2 ~ 
/sin ~ 3 x-dx = s 1" *?• 

Q , 1 + cos x 3 cot re 2 cot x 

I log + — ; + . o 

n . , 7 1 — cos re sm re sm^rc ~ 
jsnr^X'dx = 2~7 f" <?• 

n _ , 1 + cos re 15 cot re 10 cot re 8 cot re 

¥■ log + — : + . 3 + . . 

n . 7 , 1 - cos re sm re sm 3 re sm 5 re ~ 
j sin- 7 rc-drc = + C 



8 THE INTEGRAL CALCULUS 

The law of progression is 

f-^-ch = -=- f-^—dx - — L_ .25**. 
J sin m+2 a; m + 1 J sm m x m + 1 sin m z 

The value of C is the full quadrant of the curvilinear. 

4. The value assigned to C in the integrals of — dx, in 

Tables I and IV, is readily derived from the following consideration. 

If we trace, for the full quadrant, the two curves, y = — = and 

1 ' ' u sm m x 

y = — — , having their origin at opposite ends of the axis of x, the 

two curves will coincide throughout. Therefore, the value of 

I -dXj given by Tables I and IV, simply measures negatively 

the value given by Tables II and III for I -dx. Hence, 

& J J cos w z 

dx measures negatively the complement of the area of the 



/; 



sur'z 

curvilinear ; so that, adding to this negative quantity the full quad- 
rant of the curvilinear gives the proper integral. These remarks 

equally apply to the curves y = tan™£ and y = - — — , the integrals 

tan x 

of which are given on another page. 



Series I 
The integral of sm. m x-dx, for any value of m, is A-(l — cos x) 
— -(1 - cosV) + — (1 - cos 5 z) - —(1 - cosV) + . . . 
where A, B, C, D . . . are the successive terms in the development 
of the binomial (1 + 1) 2 ; namely, A = 1; B = — - — ; 
„ m — 1 • m — 3 n _ m — I'm — S-m — 5 

5. Series I terminates with the term containing the mth. power 
of cos x, when m is a positive odd integer; otherwise it is infinite. 
It therefore gives complete expressions for odd positive powers of 
sin x - dx. 

Series II 
The integral of cos™£ • eta, for any value of m, is A -sin x — — sin 3 x 

C D 

+ — sin 5 z - — sin 7 x + . . . where A, B, C, D . . . are the suc- 
5 7 

m— 1 

cessive terms of the binomial (1 + 1)~2~, as before. 



THE INTEGRAL CALCULUS 9 

This series will terminate with the rath power of sin x when ra is 
a positive odd integer; otherwise it is infinite. It, therefore, gives 
complete expressions for positive odd powers of cos x -dx. 

Series III 

6. The integral of tan w £ • dx, for even values of ra, is ± x =f tan x 

tan 3 x tan 5 x , tan m_1 a: . n . 

± — x — =f — - — ± ... to term — , using the upper or lower 

O O Til — J. 

ra 
sign according as -^ is even or odd. 

This series, as therein indicated, will terminate with the (ra — l)th 
power of tan x. 

Series IV 

7. The integral of ta,n m X'dx, for odd values of ra, is ± log cos x 

tan 2 # tan 4 x tan 6 x , , tan m-1 :r . ,, 

± —^ — =»= — z — =*= — n — ^ ... to term — ; using the upper 

2 4 o ra — 1 ^ 

yyi \ 

or lower sign according as — » — is odd or even. 

This series, as therein indicated, terminates with the (ra - l)th 
power of tan x. 

Series V 

8. The integral of cot w z • dx, for even values of ra, is 

cot 3 z cot 5 x , , cot m-1 :r ~ 

± x =*= cot x =f — ~ — ± — = — ^ ... to term — , + C, 

3 5 ra - 1 ' 

ra 
using the upper or lower sign according as ■=- is even or odd. 

The value of C is the area of the full quadrant of the curvilinear, 

7T 

* 2' 

Series VI 

9. The integral of cot™£ • dx, for odd values of ra, is =f log sin x 

cot 2 # cot 4 x cot 6 # . . cot w_1 ^ ~ . ,, 

=f —tz— =*= — -. — =»= „ ± ... to term -. + C using the up- 

2 4 6 ra - 1 ^ 

per or lower sign according as — » — is odd or even. C is the full 

quadrant of the curvilinear. Series V and VI, as therein indicated, 
terminate with the (m — l)th power of cot x. 
For the value of C, see remarks under Art. 4. 

Series VII 

10. The integral of ta,n m x-dx, for any value of ra, is — 

& J ' ra + 1 

tan w+3 x tsm m+5 x tan m+7 a: ^ T , 
— -f- = — =- + . . . When ra is an even nega- 

ra + 3 ra + 5 ra + 7 



10 THE INTEGRAL CALCULUS 

tive, the constant C to be added is the full quadrant of the curvilinear, 

± - ; using the upper or lower sign according as -jr is even or 

odd. When m is an odd negative, the constant C to be added is 
the full quadrant of the curvilinear. When m is an odd negative, 
there will always appear in the series one irrational term, namely, 

± - tan°z. It is quite permissible to suppose that this term has 

been obtained by integrating, with respect to the tangent, the 

quantity j tan -1 .?, = - — — ; but the integral of this, with respect 

_L LtxXl JO 

to the tangent, is log tan x. Therefore, log • tan x must always be 
substituted for the irrational term, using the sign + or - according 
as it is in the series. 

11. Series VII is infinite for any single integral; but it terminates 
for certain pairs of integrals, thus: 

tan w+1 z 



J % taji m x • dx + ytan w+2 a; • dx = 



m + 1 ' 



ni ,, , tan w+1 ^ tan TO+3 x tan m+5 x 

ft&n m x-dx + ftsm m+6 x-dx = — = -=- + — =-; 

"* m + 1 m + 3 m + 5 

and, r being any whole number divisible by 4, 

ni ,., , tan w+1 o: tan 7 "^ tan m+5 x 

ft2M m X-dx + ft2M m+2+r X-dx = — : — k- + — r 

° m + lm + 3m-ho 

tan « i +l+r a . 

to term =*= z — ■ — 

m + 1 + r 



SECTION TWO 

Showing how Some Well-known Mathematical Formulae, derived 
from other Sources, may be obtained from the Foregoing 
Integrals 

12. It will be observed that each of the integrals given by Tables 
I and II, for positive values of m, contains x; that is, some definite 
proportion of 7r; while tv does not enter into the values given by 
Series I and II. It is, therefore, evident that a great variety of series 
for expressing t may be had by equating the value given by the 
series to that given by the tables. A few examples are here given. 

7T 

Let x = -z , and equate Series I to Table I, for 

J^ _3_ 3-5 3-5-7 t 

m -°' then l + 2-3 + 2-4-5 + 2-4-6-7 + 2-4-6-8-9 + ' ' ~ 2 

m = 2, then 1 — 



2- 


4-6-8- 
3-5 


9 


2 


•4-6-8 
3-3 


•9 



2-3 2-4-5 2-4-6-7 2-4-6-8-9 

_3_ _3_ 3_ 

m - 4, then 2 3 + 2 4 >g + 2 . 4 . 6 . 7 + 2 . 4-6-8-9 

3-3-5 3tt 

+ 2-4-6-8-10-ll + ' ' * " 16* 

13. By equating Series II to Table II for m = 0, we get 

This series expresses the arc in terms of the sine; and it is simply 
an inversion of the series given by Newton for the sine in terms of 
the arc. 

14. By Table I, the integral of sirPx-dx is » , and 

by Series I, it is 

a _ e) _^ (1 . <f ,_ 5 ^- I i_ <f) _ 5: ^_ t i_ rf , 

3-5 

(1 — c 9 ) - ... or the equivalent series given in Art. 29, 



2-4-6-8-9 



I 03 i JL £5 . _^ .07 i 35 .09 . 3 ' 57 .Oil . 

3 "^2-5 ^2-4-7 ^2-4-6-9 ^2-4-6-8-ll t . - . 

11 



12 



THE INTEGRAL CALCULUS 



These two series express the same quantity as the integral of y/2x — x 2 } 
in the equation to the circle y = -\/2x — x 2 . (Here, x is an algebraic, 
not a transcendental quantity, although it is incidentally the versed 
sine of the arc.) This may be shown by tracing the two curves. 



Fig. l. 




Fig. 2. 




Figure 1 is a quadrant of the circle y = V2x - x 2 ; Figure 2 is 
a quadrant of the curve y = sin 2 #. 

7T 

AC = 1, ac = - = arc EBC; DC and cd = x, BD and bd = y in 
the respective equations. The areas BCD and bed are the integrals 
of y. The area bed, as shown above, is -= — ■= • 



When the arc BC = axis cd = x, then 
Area BCD 



BCA - BDA = | - cosx ^ mx = area bed. 



In Figure 2 the length of the curve for a full quadrant is the same 
as the length of the curve of sines, and the same as the ellipse whose 
semi-axes are V2 and 1. And certain modifications of the curve of 
sines, by introducing constants, give the same length of curve as 

the ellipse whose semiaxes are a and b; thus: y = Va 2 — 6 2 -sin-T 

gives the same length of curve as the ellipse, with the centre as origin 

x 



of coordinates; and y = Va 2 - 6 2 -cos-t gives the same as the 

ellipse, with the apex as origin. 

This may be shown geometrically, but the demonstration would 
be foreign to the purpose of this work. 

15. Let the complete integrals of sin m X'dx for odd values of m 
be deduced from Series I, thus: 
S sin x-dx = 1 - cos x. 
y* sin 3 z • dx = f - cos x + J cos 3 £. 
fsirfx-dx = f 'i - cos x + f cos 3 x - } cos 5 z. 

7T 

And, if the full quadrant be taken, x = -z ; therefore, 



x 



2sin ,n x-rfx = § -t-f-f -tr ... to 



m — 1 
m 



THE INTEGRAL CALCULUS 13 

Now, by Table I, for even values of m, we get 



x 



2sm m x-dx = - (4"i'£-£ ; A ... to 



2 V 2 4 6 8 10 rn 

But, if m be considered infinite, the distinction between odd and 
even values of m vanishes, and these two expressions will be equal 
to each other. Hence, if the first expression be divided by that part 
of the second which is within the parentheses, we get 

T ^2 4-4 (v6 8-8 10-10 

2 ~ i.3 , 3-5'5-7"7;9"*9-li * ' '' 

and this is identical with the series given by Wallis for the value of tt. 

16. If the integrals given by Table II for negative values of m 
be equated to those given by Series II, we get expressions for the 
tangent in terms of the sine. Thus, for ycos~ 2 #, 

sin x + \ sin 3 z + ^— -. sin 5 # + . a sin 7 # + . , sin 9 x + . . . 
2-4 2-4-6 2-4-6-8 

= tan x. 

And this is the series that results from dividing the sine by the 

sin x 
cosine; that is — , the denominator being reduced to a series, 

v 1 — sin 2 # 

17. For any even value of m, if the integral of tsu^x-dx given by 
Series VII be equated to that given by Series III when m is positive, 
or to that given by Series V when m is negative, the result in each 
case will be, 

x = tan x - \ tan 3 # + i tan 5 # - \ tan 7 # + . . . 

and this is Gregory's well-known series for expressing the arc in 

terms of the tangent. 

The separate values of the positive and negative parts of this 

series may be had by comparing it with the integral of dx, given 

cos X 

by Series II, thus; 

The positive part = J log z + h x - 

l — tan x 

The negative part = \ log -i \x. 

1 — tan x 

This is true only when the tangent is less than 1, because, when 
it is 1 or greater than 1, each part is infinite. 

18. For any odd value of m, if the integral of ta,n m x -dx given by 
Series VII be equated to that given by Series IV when m is positive, 



14 THE INTEGRAL CALCULUS 

or to that given by Series VI when m is negative, the result will be 
| tan*x - J tan 4 # + ^ tan 6 z - f tan 8 z + . . . = - log cos x, 
or its equivalent, 

i cot 2 x - i cot 4 z + | cot 6 £ - J cot 8 x +...=- log sin x. 
Making tan x = 1, and multiplying the result by 2, we get 

i - i + i - i + * - i + ■ • • = log 2. 

Making tan x = §, we get 

2(4) 2 ^3(4) 3 4(4)< 
If a and 2 be any positive quantities, will represent a co- 
sine in the first quadrant, and its corresponding tangent will be 



2 + ^TTVs - TTTu + • • • = log 5 - log 4. 



\/2az + z 2 

Using this value of the tangent, we get a series for ex- 
pressing the difference between log a and log (a + z) ; 

2az + z 2 (2az + z 2 ) 2 (2az + z 2 ) 3 . , . _ 

~W 4^— + 6a° ~ ' * * = log (a + z) - log a. 

And, making z = 1, and a = the consecutive numbers in successive 
operations, we get a series for constructing tables of logarithms, thus: 

2a + 1 (2a + l) 2 _ J (2a + l) 3 (2a + l) 4 _ j - 

^a^-"la^ + ^6a^^-^^^ + ■■-«•** («+l ) 

- log a. 



Again, making the cosine , its tangent will be — -• Using 

v a + z Va 

this value of the tangent, and multiplying the result by 2, we get, 

And this is a well-known series for constructing tables of logarithms. 
In this series, z may be considered negative as well as positive, by 
placing the tangent in the second quadrant; hence, we get, 

l + £* + & + & + ■ ■ ■ = l°g «" log («-*), 

and, taking the sum and difference of these two series, we have 

2 (a + & + £ + 6 + • • • ) = l0g {a + Z) ~ l0g (a - 2); 
and 

2 (£* + & + & + & + ■ ■ )=log« 2 -log(a 4 -^. 



THE INTEGRAL CALCULUS 15 

1 

19. By Series II. for the integral of dx, we get, sin x + 4- sin 3 £ 

cos x a 



+ ^ sin 5 £ + . . . which is the same in form as the series given above 

r hich case, 
1 + sin x 



for log (a + z) — log (a - z), when we make sin x = — In which case, 

tit 



the sum of this series is \ log (a + z) — \ log (a — z) = | log 



1 - sin x' 



and this agrees with the value of I dx, given by Table III. 

J cos x J 



SECTION THREE 

On the Series for expressing the Tangent in Terms of the Arc, and 
its Relation to the Bernoullian Numbers. 

20. The application of Maclaurin's theorem of successive differ- 
entials, to express the tangent in terms of the arc, is generally con- 
sidered too troublesome for practical use on account of the com- 
plexity of the derived functions. I offer a solution of this problem, 
in which the difficulty is obviated by applying Maclaurin's theorem 
to the equation f tan x-dx = - log cos x } given by Series IV. 

Assume 
w = - log cos x = Ax + Bx 2 + Cx z + Dx 4 + Ex 5 + . . . (1) 

^ = tan x = A + 2Bx + SCx 2 + 4Dx* + . . . (2) 

ax 

5-? = 1 + tan 2 z = 2B + 2-3Cz + 3-4DZ 2 + 4-5#z 3 + . . . hence 
ax 2 

tan 2 x = -1 + 25 + 2-ZCx + 3-4Dz 2 + 4-5#z 3 + . . . (3) 

By taking the square of series (2) for tan x, and equating its 

coefficients for like powers of x to those of series (3) for tan 2 z, we get, 

A,C,E,G,I . . . each = 0. 

/? i D l F l TT 17 T 62 



3-4 5-9 5-7-8-9 5-7-9-9-10 ' 

Raise these fractions so as to have an obvious progression in the 
denominators, and substitute them for B, D, F, H, . . . in series (2) 
and we get, 

2 3 16 s 272 7 

tans -z + 23 x +2.3.4.5* + 2-3.4-5-6-7* 

7936 9 r . 

Here, the numerators of the coefficients cannot be determined by 
inspection, and it required much thought to evolve their law of 
progression, which is as follows: 

21. Let iVi, iV 3 , Ns, JVV . . . represent the numerators of the 
coefficients of x, X s , x 5 , x 7 . . .; and let a represent the index of the 
power of x whose coefficient is required; then the numerator, N a , 
of the coefficient x a is 

a«a — l*a — 2,, a-a - 1-a - 2-a - 3-a - 4 , T 
*l*aY,-. 24--*^- 2-3.4-5 -*.*■■■ 

16 



THE INTEGRAL CALCULUS 17 

to the term N a — 2- Using the upper or lower sign according as 
— - — is even or odd. 

A table of these numerators is given at the end of this section. 

If the values of B, D, F, H . . . be applied to series (1) Art. 20, 
we get 
_1 _i 2 2 4 16 6 272 

iogcosx- 2 z + 2 . 3 . 4 * + 2 . 3-4-5. 6^ + 2.3-4-5-6-7-8* + '" 

which is simply Series (4), Art. 20, integrated with respect to x. 

22. The above law for the numerators of coefficients may be 
modified so as to apply to even powers of x, thus, 

a-a — 1 , T a-a — \-a — 2-a — 3 , T 

N 2 =*= 7^—. N 4 



2 " 2-3-4 

a-a — 1-a — 2-a — 3 -a — 4 -a — 5 



N* 



2-3-4-5-6 
to the term N a —2- Using the upper or lower sign according as 

s is odd or even. From this we get 

1 i 1 2 4 6 looO p 

+ * X + 2^4* + 2-3-4-5-6 X + 2^3-4-5-6-7-8 

50521 10 

+ 2-3-4-5-6-7-8-9-10 a; + • • • C 1 ) 

Now, the square of this series is 

i , 16 272 . 7936 

1 + X + 2^i X ' + 2^Q X + 2^8 XS+ ' ' * (2) 

But this is the differential with respect to x, of series (4), Art. 20; 

namely, 1 + tan 2 x. Therefore, Series (1) = Vl + tan 2 :r = secant x. 
By integrating Series (1) with respect to x, we get 

r A 3 i 5 7 looO 9 

J sec x • ax = x + ~ __ x -j- _ „ . — # -f- — ^ £ + — - ic + . . . 

It is worth noting the variety of expressions we have for J^sec x • dx. 
Here it is given in terms of the arc. By Table III, making m = 1, we 
have it in terms of the logarithm of the sine; by Series II, making 
m = - 1, we have it in terms of the sine; by Art. 30, making m = — 1, 
we have it in terms of the cosine; by Art. 33, making m - 1, we 
have it in terms of the secant; and by Art. 39, it is the solving ele- 
ment in the rectification of the logarithmic curve. 

The integral of secx-dx, by Art. 33, is log (sec-x + tsm-x); let this 

= z = x + t^-^ x 3 + » . g x 5 + s = x 7 + ... as above. Now, 

2-3 2-3-4-5 2- • -7 



18 THE INTEGRAL CALCULUS 

when this series is reversed, it has the same coefficients, with the 
signs alternately + and — ; 

1 3 5 5 61 7 

x ~ Z 2-Z Z + 2-3-4-5 2 2---7* + * * * 

And e z — e~ z = 2 tan • x ; e z + e~ z = 2 • sec • x. 

23. The series commonly given in mathematical works, for express- 
ing the tangent in terms of the arc, is derived from Euler's formulae 
for the sine and cosine. Let i = V- 1, then these formulae are: 

sinz = prr (e xi - e~ xi ). 
2% 

cosz = \{e xi + e-™). 
And if the first be divided by the second, we get, 

tan x = -1 1 — 



i\ 1 + e 2xl , 

Now, if this last expression be expanded in a series, the result 
will be identical with series (4), Art. 20. Mathematicans, however, 
have not given the result in that form, evidently because of the 
difficulty with the law or coefficients; but they have given this 
equivalent form, 

tana: = 2*(2 2 - l)^ + 2<(2* - 1) ^ + 2«(2 6 - D 2.3^5,6 + : ' ' 

where Bi, B 2 , Bz, J5 4 . . . are the celebrated Bernoullian Numbers. 

The relation between these numbers and the numerators of 
coefficients is readily established by comparing the two series in 
which they respectively occur, namely: 

■kt a + 1 „ 

24. Series (4), Art. 20, may be derived from another source, more 
elementary than the foregoing. 

Take the simple formula in trigonometry, 

, 7 N tan a + tan b 

tan(a + b) = r 

1 — tan a • tan b 

and let b = a, 2a, 3a . . . ma, in successive operations; then 

2 tan a 3 tan a - tan 3 a 

tan 2a = 1 - tanV tan 3a = 1 -Stanza ' 

4 tan a - 4 tan 3 a _ 5 tan a - 10 tan 3 a + tan 5 a 

tan 4a = ^— — ^ r — r-, tan 5a = 



1-6 tan 2 a + tan 4 a' 1-10 tan 2 a + 5 tan 4 a 

Here, the law of progression is this: for tan (ma), express 
(1 + tan a) m in a series, and place the successive terms in the de- 
nominator and in the numerator alternately, and change the signs 



THE INTEGRAL CALCULUS 19 

so \hat they will be + and - alternately, in the denominator and 
also in the numerator. Now, in these expressions for multiple tan- 
it 
gents, let us substitute arc - for arc a. Let x represent any propor- 

x 
tion of ir, and let n be infinitely large. Then, tan -, being infinitely 

small, does not differ from its arc, -. Hence, these multiple tangents 

n 



may be written, 






A A 


V n) ~ l _ z_ 2 ' Un V n) ~ x _ 3 t' 

n 2 n 2 


"VnJ x 2 x v 
1 - 6-£ + ~I 


and, generally, tanf m - J = 




a: m-m — 1 • m — 2 # 3 m-m — 1-m 


— 2-m — 3-m — 4 x 5 


111 n ' 2-3 n 3 ' 2-3-4-5 ra 5 " 


1 m-m — 1 .x 2 , m-m— 1-m— 2-m— 3 x 4 


m-m — l . . . m— 5 a: 6 
2-3-4-5-6 n 6+ " 


2 n 2 ' 2-3-4 n A 



Since this is true for every value of m, let m be considered infinitely 
large. Then m — 1, m - 2, m - 3 . . . each equals m. Now, in each 
term of the series, m has the same power in the numerator of the coef- 
ficient as n has in the denominator of the argument; and, both being 
infinite, they unitize each other throughout, in both sides of the 
equation; and the series is, therefore, reduced to this, 

2-3 2-3-4-5 2-3-4-5-6-7 + ' ' ' 
tan x = = 

1 _ — r 2 4- r 4 — r 6 -I- 

2 + 2-3-4 2-3-4-5-6 + ' ' ' 

But, since x represents any proportion of ir, the numerator of this 
series expresses the sine, and the denominator the cosine, in terms 
of the arc; and if the one be divided by the other, the resulting 
series will be identical with Series (4), Art. 20. Therefore, this single 
operation on the theory of multiple tangents gives series for the sine, 
the cosine and the tangent, in terms of the arc. 

On the Catenation of the Bernoullian Numbers 
25. Definition. If the sum of the series, l p + 2 P + 3 P + 4 P + 
. . . n p , be expressed in terms of the powers of n, the coefficients of 
n l in these expressions, for even values of p, will be the Bernoullian 
Numbers; p = 2 giving J5i; p = 4 giving B 2 , etc. The sum of 
this series may be found in terms of n by the Theory of Differences, 



20 



THE INTEGRAL CALCULUS 



as follows : Find the successive Orders of Differences in the series 
a, b, c, d, e . . ., thus: 



ai 


1st 


2d Order 




3d Order 




4 th Order 


5th Order 




Order 














GO 


dv 


d 2 




d 3 




di 


do 


a 
















b 


b — a 














c 


c-b 


c - 2b + a 












d 


d - c 


d -2c + b 


d- 


3c + 36 - a 








e 


e - d 


e - 2d + c 


e - 


3d + 3c -b 


c - 


- M + 6c - 46 + a 




f 


f-c 


f -2e + d 


f- 


■ 3e + 3d - c 


f- 


- 4e + 6d - 4c + b 


/ - 5e + lOd 

- 10c + 56 - a 



Here, the First Difference of the rath Order of Differences, d m , is 



mb 



m-rn — 1 



ra-ra — 1-ra — 2 



d 



2 2-3 

using the upper or lower sign according as ra is odd or even. And 
the sum of the series a + b + c + d+ . . . n is 



n-n — 1 7 n-n — 1-n 
na -\ ^ ai + 



2 7 n-n — 1-n — 2-n — 3 7 
d 2 + ■ <-, , «3 -f • • . , 



2 ~ x ' 2-3 ~ 4 ' 2-3-4 

where d h d 2 , d 3 . . . are the First Differences in the respective Orders 
of Differences. 

For a + b + c + d + . . ., substitute the series, l p + 2 P + 3 P + 4? 

+ . . . n p = £. 

Let p = 1, then $ = ^n 2 + -Jn. 



n „ „ n-n — 1 , n-n 
Let p = 2, then >S = n H = «i + 



l-n-2 , 1 . 

— di = - n a 



2-3 



1 , 1 
+ 2 n+ 6 n - 



The coefficient of n 1 , here, is ^; therefore £>i = J. 

„ ., „ n-n— 1 7 n-n — 1-n — 2, 
Let p = 3, then £ = n H -= c/i + ^To "2 

n-n— 1-n — 2-n — 3 7 1 . , 1 „ , 1 „ 
+ 2^4 A^ + gn' + jn*. 

Let p = 4, then £ = ^n 5 + |n 4 + §n 3 - ^n. 

The coefficient of n 1 in this case is ¥ V; therefore Z? 2 = gV- 

The successive Bernoullian Numbers may be found as above, but 
the labor increases very rapidly with the higher values of p. 

Now, collect the foregoing results into one view, together with 
others derived from the same source, thus: 

p - 1, S - in 2 + in. 

p = 2, S = in 3 -4- Jn 2 + Jn. 
p = 3, 5 ■ i n 4 + in 3 + in 2 . 



THE INTEGRAL CALCULUS 21 

p = 4, S = in 5 + \n 4 + |n 3 - -fan. 

p = 5, S =. Jn 6 + §n 5 + T %n 4 - T \n 2 . 

p =-6, S = }n 7 + in 6 4- \n h - |n 3 + -fan. 

p = 7, S = |n 8 + |^ 7 + i^n 6 - ^n 4 + ^n 2 . 

p = 8, £ = ^n 9 + Jn 8 + f^ 7 - ts^ 5 + f ™ 3 - ik n - 

p = 9, S = ^n w + |n 9 + f n 8 - -^n 6 + in 4 - -^nK 

In each of these values of S, it is evident that the 

First term is 7 N p+l . 

p + 1 

Second term is %N P . 

Third term is | • ^ iV*" 1 , = | BiA^" 1 . ' 

Fourth term is ^ ^ ^ 30 ' = 2-3-4 2 

Fifth term is *JL 2-3-4-5-6 42 ^ ' 

2-3-4-5-6 3 * 

Hence, we get the general equation, l p + 2 P + 3 P 4- 4 P + . . . n p 

AT P +i + I N p + 2 B^" 1 - P ' y ~ 1 o P ,~ 2 # 2 A^- 3 



p + 1 ' 2 2 x 2-3-4 

p-p — 1 . . . p — 4 „ , T B p-p — 1 . . ■. '» — 6 „ , T , 

+ 2-3 ... 6 3 2-3 ... 8 4 + • • -i 

omitting the term N°, which occurs when p is an odd number. Since 
this equation is true for all cases where p and n are positive integers, 
it is true when n = 1 ; that is, when the numerical side of the equa- 
tion is reduced to the first term, namely, l p . Making n = 1, we get 
the equation 

1= ^n + 2 + 2 5l ~ 2-3-4 Bi 

p-p - \-p -2-p -3-p -4 
" + " 2-3-4-5-6 3 

This series always terminates with the term Bv. These Numbers 

2 
may, therefore, be directly derived from it with greater facility than 

from the Theory of Differences; although that theory was neces- 
sary to establish the series. 

Let p = 2, then 1 = \ + J + B x . Therefore B x = f . 
p = 4, then 1 = ± + \ + f - B 2 . :. B 2 = ^. 
p = 6, then 1 = } 4 \ 4 $ - <& + B 3 . .'. B 3 = A- 
p = 8, then 1 = i 4- i + * - iJ + I - B4. /. £ 4 = A- 
p = 10, then 1 = T V + § + f - U + if - M + B 6 . .'. 5 5 = eV 



22 



THE INTEGRAL CALCULUS 



26. The first seventeen of these Numbers are given in the follow- 
ing table, together with the Numerators of Coefficients. They were 
calculated by this last series, and verified by an independent calcu- 
lation of the Numerators by the law given in Art. 21 and 22. 

Numerators of Coefficients for Odd and Even Powers of X, and the 
Bernoullian Numbers Corresponding with the Odd Powers of X 



X° 

x° 

X 1 






Na 






1 
1 


Bi 


B a+l 
2 

1/6 


X 2 

X 3 












1 

2 


B 2 


1/30 


X 4 
X 5 












5 
16 


#3 


1/42 


X 6 
X 7 












61 

272 


B 4 


1/30 


X 8 
X 9 


• 










1385 
7936 


B 5 


5/66 


X 10 
X 11 










3 


50521 
53792 


Be 


691/2730 


X 12 
X 13 










27 
223 


02765 

68256 


B 7 


7/6 


X 14 
X 15 










1993 
19037 


60981 
57312 


B s 


3617/510 


X 16 
X 17 








1 
20 


93915 
98653 


12145 
42976 


B, 


43867/798 


X 18 
X 19 








240 

2908 


48796 

88851 


75441 
12832 


Bio 


1 74611/330 


X 20 
X 21 






37037 
4 95149 


11882 
80531 


37525 
24096 


B n 


8 54513/138 


X 22 
X 23 






69 
1015 


34887 
42388 


43931 

65068 


37901 
52352 


Bn 


2363 64091/3730 


X 24 
X 25 




2 


15514 
46921 


53416 
48019 


35570 
02079 


86905 
83616 


B\z 


85 53103/6 


X 2G 
X 27 




40 
702 


87072 
51601 


50929 
60394 


31238 
39598 


92361 

87872 


Bu 


2 37494 61029/870 


X 28 
X 29 


2 


12522 
31191 


59461 

84187 


40362 
80959 


98654 
78414 


68285 
73536 


Bin 


861 58412 76005/14322 


X 30 
X 31 


44 

871 


1543S 
39627 


93249 
57125 


02310 
16929 


45536 
61708 


82821 
11392 


Bi& 


770 93210 41217/510 


X 32 
X 33 


17751 
3 72940 


93915 
77037 


79539 
20529 


28943 
57109 


66647 
75096 


89665 
25856 


Byj 


257 76878 58367/6 



SECTION FOUR 

Theorem for integrating the Powers of a Transcendental Function, 
in Terms of the Function considered as an Algebraic Quantity 

27. In Series I and II, it will be observed, the integrals of sin m x-dx 
are expressed in terms of cos x; and those for cos m x-dx are expressed 
in terms of sin x. In my efforts to express these functions in their 
own terms, I found the following method to be of general application : 

dy 
Let y = 4>x. Divide y by -j- expressed in terms of y; 

thus, , x ; raise y to the rath power and we have -~ 



fdy\ 
\dxj 



Integrate this last expression, with respect to y, as an independent 
variable and as an algebraic function. The result will be the integral 
of (cj)x) m -dx, expressed in terms of the function itself, whether it 
be algebraic or transcendental. 

dy y m 

28. Let y = x 3 . — = 3x 2 = 3y*. Then ~-^ integrated with respect 

to y = ^-~- jv Therefore, the integral of (x s ) m 



3(ra + 4) ' & v J ~3ra + l 

dy ym 

29. Let y = sin x. -j- = cos x = Vl - y 2 . Then — being 

reduced to a series, and integrated with respect to y, gives: 

fsin m x • dx = — i— S m+1 + ' \ £-+ 3 + — — £-+ 5 

ra + 1 2(ra + 3) 2-4(ra + 5) 

3-5 

+ o a at r~^\ S m+7 +.-.., where S represents sin x. 

2-4-6(ra + 7) 

When ra is negative, the constant C to be added is the full quad- 
rant of the curvilinear for the particular value of ra. Giving ra par- 
ticular values, we have 

f&mx-dx = 2 S2 + 2^ Sl + 2^H> Se + 2 -4-6-8 * S8 +'"= * ~ cosx - 
S**x-d* - l& + ±S> + ^ + ^S» +• • • . 

x — sin x • cos x 
= 2 

23 



24 THE INTEGRAL CALCULUS 

dy 



30. Let y = cos x • -£- = — sin x = — Vl — 2/ 2 . Then 



integrated with respect to 2/, gives 

cos w+1 rc cos m+3 re 3 cos m+5 rc 3 • 5 cos m+7 rc 

m + 1 + 2(m +3) + 2-4(m + 5) + 2-4-6(ra + 7) + " 

This requires the constant C to be added, namely, the full quad- 
rant of the curvilinear. But the full quadrant is the same for sin m rc 
as it is for cos m rc; and for sin w rc the integral is 

^w+l Jm+3 3.Jm+5 3'5'l m + 7 

+ n , ■ , ox + T^rs t^t + n * n , — t^t + * • • 



m + 1 ' 2(m"+3) 2-4- (m + 5) 2-4-6- (m + 7) 
Now, adding this to the above series, we get 

(1 - c m+l ) (1 - c m+3 ) 3(1- c™+ 5 ) 3-5(1- c m + 7 ) 
fcos m x-dx = v 



m + 1 ' 2(m + 3) ' 2-4(m + 5) ' 2-4-6(m + 7) ' ' 

where c represents cos x. This series eliminates the constant. 

When m is an odd negative, these series, will have an irrational 
term. In Art. 29, substitute for it log sin x; and in Art. 30, substi- 
tute for it (1 -log cos re); retaining in each case the coefficient 
given by the series, excepting that the zero in the denominator is 
to be disregarded. The sum of these series may always be had from 
Tables I to IV, when m is a whole number, positive or negative. 

dv 
31. Let y = tan x. -j- = 1 + t 2 = 1 + y 2 . And the integral of 

n.m t m ^ fm+Z pn+5 



= rtsm m x = — r — r + 



1+2/ 2 * m + 1 m + 3 m + 5 

This is identical with Series VII, and, therefore, the same condi- 
tions prevail as to the irrational term, constants and negative values. 



32. Let y = V2x - x 2 . Here x is an algebraic quantity; and 
this is the equation to the circle independently of trigonometrical 
functions; but its integral, nevertheless, will be the integral of the 
sine with respect to the versed sine of the arc. 



dy 1 - x 1 - x Vl - y 2 m , y m y m+l 

dx ~ V2z - x 2 y y VT^Y 2 " vi - y 2 

y 

Substituting S for y, the integral of this is, 

f sm m rc-a-ver x = — = + ^ — tt + 



m + 2 2(m + 4) 2-4 (m + 6) 

3-5 S m+8 
+ 2-4-6(m + 8) + '" 



THE INTEGRAL CALCULUS 25 

Giving particular values to m, 

m = 0; -*S 2 + ^— - S 4 + n . _ & 6 + . . £ 8 H = versed sine of arc 

- 2-4 2 • 4 • b 2 • 4 • o • 8 

of which S is the 

sine. 

m x '3 + 2-5 + 2-4-7 + 2-4-6-9 + ~ 2 

m = 2;^ + ^^ + ^3 + _^^o + ... =ver2 _l ver , 



Taking the similar case, ?/ = — s/2ax - x 2 , the equation to the ellipse; 
and omitting the constant - for the present, we have, 



dy a — x a - x Va 2 - y 2 Th b f* y m + 



s. 



dx ^/2ax - x 2 V V a J Va 2 - y 2 

b f y mJr ^ ym+4 Qym+6 Q . C ytn+S 

I C\ O / ■ A \ ~' C\ A A / i /"»\ 



a 2 \m + 2 ' 2a 2 (m + 4) ' 2-4a 4 (m + 6) ' 2-4-6a 6 (m + 8) 



Here, y = V2ax — x 2 . 
Making m = 1, we get, 



bfl 3 _1_ 5 _3 7 3-5 

a2 \^y + 2 -5a 2?/ + 2-4-7a 4?/ + 2-4-6-9a 6 



This measures the area of the ellipse from the apex as origin. There- 

y 

fore, if for any ordinate y, we make - = sm v, then the area of the 



ellipse from the apex to that ordinate is (a -b) 



a 

v — sin v • cos v 



2 

If the full quadrant be considered, then x = a = y, and we get, 

(1 J_ _3_ 3-5 

U + 2-5 + 2-4-7 + 2-4-6-9 + " 

The numerical series here is the same as in Art. 29, when m = 2 and 

IT IT 

S = 1. It, therefore, equals j and a-b-j is a quadrant of the 

ellipse whose semi-axes are a and b. 

When x is greater than a, the series measures the complement of 
the second quadrant; y being a maximum when x = a. 



26 THE INTEGRAL CALCULUS 




33. Let y = sec x = vT+7 2 - -f- = f = y-Vy 2 - 1. 



Then = ; and the integral is 

?/-Vi/ 2 - 1 VT - 1 

J-i J- o O e O'O -, 

m-l y ^2(m-S) y ^2-4(m-5)^ ^2-4-6(m-7r 

+ ••• 
Giving particular values to ???, 
??z = 1 ; then 7" sec x-dx = 

1 3 3-5 3-5-7 

g?/ 2-2?/ 2 2-4-4?/ 4 2-4-G-6?/ 6 2-4-G-8-8?/ 8 

Here, when x = 0, the area = 0, and y = 1; therefore logy = 0. 
Hence, the constant, to be added, is the negative part of the series 
taken positively. Therefore, C = + log 2. But, for any value of x } 
the negative part of the series is log [l + (Vl + t 2 - t) 2 '] = log 2(1 + t 2 
-t-VT+T 2 )- 

Therefore, 



/sec x-dx = log Vl+t 2 - log (1 + t 2 - tVTTT 2 ) = log(vT+7 2 + 0- 

Hence, the constant is eliminated. 
Let m = 3, then 

n ' 1,1, 3 3-5 3-5-7 

fsec*X'dx = -y 2 + -log?/ - 



2* ' 2 fe * 2-4-2?/ 2 2-4-6-4^ 2-4-6-8-6z/ 6 " 
The constant C, to be added, is J log 2 - J. 
But, for odd values of ??i, 

m r 1 

fsec m+2 X'dx = I sec m x-dx -\ ; -tan x-sec m x. 

^ m + 1 J m + 1 

Hence, 

fsec 3 x-dx = ilogVT+T 2 + it-VTTT 2 - i log(l + t 2 - t-VTT?- ) 



= Jlog(vT+i 2 + +it-VTTi T . 

The sum of the series for other values of m may be had from 

Tables II and III and Series II, observing that sec m x = 

cos w x 

34. On the integration of the functions x = log y, and y = log x. 
In this case, the two curves are the same, but the areas of their curvi- 
linears are different; the first curve being concave, and the second 
convex, to the axis. 

Fig. 3 represents the two curves, with A as origin of coordinates 
in each. In the equation x = log y, AE = x, positive; AF = x, 
negative; ED = y. In the equation y = log x, AC = x; CD = y; 
AB = 1. y is therefore negative when x is less than 1. 



THE INTEGRAL CALCULUS 



27 



% The two curves coincide throughout; and (omitting c, which 
represents the area for negative values of the functions) the sum of 




the areas a and b is always equal to the parallelogram whose sides 
are x and y; that is = z-logz, or its equal, y-logy. And, in like 
manner, if the two curves y = log w # and x = log™?/, be traced, the 
sum of the corresponding areas a and b will be the parallelogram 
#-log m £, or its equal y-\og m y. 



35. Let y = log x. -~ = — 

ax x 



JL 
dy 

dx 



y • x. But x = e v . 



Therefore f\og m x-dx = fy m -e y 
1 1 .. 1 



m + 1 



y 



rn+l 



+ 



+ 



m + 2 
1 



y 



m+2 



+ 



2(w + 3) 



/m+3 



i/ m+ ' 5 + 



2-3(m + 4) 



2/ 



TO+4 



2/ 



771+5 



+ 



2-3-4(ra + 5) 

This series measures the curvilinear area from the point B, towards 
the left when x is greater than 1, and towards the right when x is 
less than 1. 

The sum of this series for particular values of m, is 
yiog x • dx = x- log x - x + 1. 
flo^x-dx = z-log 2 # - 2x -log x + 2x — 2. 
J*\og z X'dx = #-log 3 £ — 3x-\og 2 x + 6x-log x — Qx + 6. 
flog^x-dx = z-log 4 x — 4a;'log 3 :r + 12x-log 2 x — 24x-log x + 24x — 24. 



28 THE INTEGRAL CALCULUS 

And, generally, making 1-2-3 -4-5 — m = q, 

flog m x-dx = ± q(x-logx - x + 1) =f |z-log 2 x ± ^"qX-log 3 ^ 

=*= 0.3.4 * l0g4x ± ' " t0 term * " ^°° mx ' usin S the upper or lower 
sign according as m is odd or even. 

36. In the equation x = log" 1 ?/, we get \/x = log y. (1) 

Therefore, y = e ax , m ~ 1 

V m—1 _ 

Hence, -j- = m-x » = 7??(vV) m_1 = m'log m ~ h y (1) 

dx 

The required integral, therefore, is mS\og m ~ l y. This measures the 
area b, (Fig. 3), from the point A, when y is greater than 1; and 
the area c, when y is less than 1. Giving particular values to m, 
and supplying the constants, we get the area for 

m = 1. y - 1. 

m = 2. 2i/-log 1/ -2y + 2. 

m = 3. 3z/-log 2 z/ - 6?/ -log 1/ + 6z/ - 6. 

And, making q = the product of the consecutive numbers from 1 to 

m, the area for ,/*log m ?/- dy = =f 5(2/ -log ?/ — y + 1) ± ^z/ - log 2 ?/ 

y ■ log 3 ?/ ± • • • to the term m • ?/ • log" 1-1 ?/, using the upper or 



2-3 

lower sign according as m is odd or even. 

If the integral given here for any particular value of m be added 
to that given in Art. 35, the sum will be the area of the parallelogram 
x-\og m x, or y-log m y. 

Wherever logarithms are mentioned throughout this work, the 
natural logarithms are intended; that is, the system whose base is 
e = 2-718281828- • • But the logarithmic integrals given in tins 
and the preceding article will equally apply to any other system of 
logarithms, if the complete expressions for these integrals be multi- 
plied by the mth. power of the modulus of that system, thus : 

Let a be the base of any system of logarithms and M the modulus 
of that system; then the integrals given in Art. 35 become 
f\og a X'dx = M{x-\og e x - x + 1) = x-\og a x — Mx + M. 
f\og a 2 x-dx = M 2 (x-\oge 2 x — 2x • log € x + 2x — 2) = X'\og a 2 X 
- 2M-x-\og a -x + 2M 2 x - 2M 2 , etc. 
And in Art. 36, the integrals become M-y — M. 

M 2 (2y\og e y - 2y + 2) = 2M-y\og a y - 2M 2 y + 2M 2 , etc. 



THE INTEGRAL CALCULUS 29 

It may be observed that the constant quantity in each of the 
integrals given in articles 35 and 36 represents the whole area (c) 
between the curve and its asymptote A F; and it is always the 
product of the consecutive numbers from 1 to m, = 1 • 2 • 3 • 4 • • • m, 
= q (multiplied by the rath power of the modulus of the system). 

Fence, between the limits of x = and 1, we have 
f *\og m -x-dx = 1-2-3-4- • -ra, 

Rectification of Curves 

37. In the foregoing we have applied the theorem under con- 
sideration only to the quadrature of curves; but it may also be 
applied with advantage to the rectification of curves. It is well 
known that, from the equation y = (fix, a secondary equation may 



vMl)'' 



be derived, namely : y = \/ 1 + -r- ; and the area of the curvi- 



linear in this second equation will be equal to the length of the 
curve in the first equation. 

To determine the length of the curve, y = x 2 , we may integrate 
by the theorem, the equation y = y/1 + 4z 2 . 

Then* ■ j!^.***- -*- * 



JL 
dy 


y 2 


dx 






3-5 



dx \/l + 4z 2 ' dy 4x 2y/ y 2 _ i 

The integral of this is 

1 2 l } 3 3-5 3-5-7 

4 2/ + 4 log?/ 4.4.2.^2 4.4.6-4-?/ 4 4-4-6-8-6?/ 6 "" W 

The sum of this series may be had by comparing it with the series 
given in Art. 33, for the integral of sec 3 x-dx, namely: 

1 2 4.li _!_ 3-5 . 3-5-7 

2 y + 2 gy 2-4-2?/ 2 2-4-6-4-I/ 4 2-4-6-8-6-?/ 6 '"' { } 

If 2x in Series (1) be considered equal to t in Series (2), then the 
sum of Series (1) will be half the sum of Series (2). Hence, we get 
the sum of Series (1) = 

1 Vl + 4# 2 + %x 1 

= - log — + 7 zX-Vl +4x 2 , or, by reducing, 

o V 1 + 4x 2 - 2x * 

= JlogCVl +4x 2 + 2x) +\xWl + 4x 2 - 
4 -« 

Now, when a; = 0, the curve is zero, and this quantity is also zero. 
The constant C is therefore zero. And the length of the curve in 
the equation y = x 2 is 



J log(\/l + 4x 2 + 2x) + \x- Vl+ 4a; 2 , 



30 



THE INTEGRAL CALCULUS 



and by introducing the constant A. the primary equation will be 
y = Ax 2 . Then the length of the curve is 



4.4 



•log- (Vl + 4AV + 2Ax) + \xWl + 4A 2 x 2 . 



38. The curve in the preceding case is the same as the curve of 
the parabola whose parameter is 1. The equation to that particular 

parabola is y = \/x. The two curvilinears form the parallelogram 
whose sides are x and y; and the two curves coincide throughout. 







D 


C 








{~1 


Fig. 4. 




A 




1 


3 


l 1 




E 



In Fig. 4, AB = x, BD = y, in the equation y - x 2 . 

AC = x, CD = y, in the equation y = ^/4ax; AE = a = \. 

Since BD = AB 2 :.DC = VAC = V^x~- 

That a similar relation between the two curves prevails for other 

values of the parameter, may be shown thus: y = V^ax, :.x = f - - 
Applying this form to the equation y = x 2 , we get y = t~; which 



differs from Art. 37 only by the constant 



4a 



From this we get 



the secondary equation ^ = yi+^z 2 ; the integral of which 
(by repeating the operation in Art. 37) is 



2a =. 



1 

+ 2 X 



^V 1+ ST 2 = a * k \ 



x\ X 



4a 2 2a/ 4a 



(0+5+s) 



i + 



4a 2 



Here, AG is the axis of x. This measures the length of the curve in 
any common parabola, where a represents the distance from the 
directrix to the principal vertex. 



THE INTEGRAL CALCULUS 31 

From the foregoing may be derived the law that Projectiles move 
in parabolic curves. Let the moving body receive its momentum 

x 
at A, in the direction AG; and let „— — represent the time, the dura- 

tion of the motion. The force of gravity deflects the body from 
the line AG towards the point D. The deflection is, therefore, BD. 

But BD = — • Hence, the deflection is in proportion to the square 

of the time. 

39. To determine the length of the logarithmic curve. In Fig. 3, 
the curve cuts the axis of x at the point B. Let this be considered 
the point of origin of coordinates; and the equation to the curve 
will be y = log (a; + 1). But the equation x = log-?/ gives the same 
curve, when EF is the axis of x, and A is the point of origin. 

dy 
Therefore, in the equation x = log-?/, y = e x , and ~- = e x . 

Then the secondary equation, y = V/l + f — ) 2 becomes y = Vl -+■ e 2x . 

Here, -y- = ; and -y- = \- = - y =1 +—+ — + — _| 

' dx Vl + e 2 * dy e 2x y 2 - 1 y 2 y* y 6 

dx. 
And the integral of this is 

1 J_ J_ J_ m 

y y 3y 3 by h 7y 7 '" Uj 

Now, by Series II, by making m = —1, we have 

8 + Is* + \S* + U? +- - - = hog 1 + SmX - (2) 



/. 



cos x 3 5 7 2 1 — sin x 

Here, if we make sin x = -, which is always possible, since 



y ( = Vl + e 2x ) is always greater than 1 ; then Series (2) will be 

equal to the negative part of Series (1). 
Hence, the sum of Series (1) is, 

Substitute for y, the differential of the curve in the equation y 



= log (x + 1), namely: Vl + e 2y = Vl + (1 + x) 2 , and we get 



1 , Vl + (1 + x) 2 + 1 



VI + (1 + x) 2 - - log v ' - + C. 

* Vl + (1 +x 2 - 1 



32 THE INTEGRAL CALCULUS 

Now, when x = 0, the curve is zero. Therefore, 

C = -V2+hog ^±l + ] = - V2 + log (V2 + 1). 

^ V2 + 1 - 1 

The length of the logarithmic curve is, therefore, 

vi + (i + xy - \ bg v * + * ; + *f 2 + ; - V2 + log ( V2 + 1) 

2 VI + (1 + x) 2 - 1 
= VI + (1 + xy - log. VI + (1 + *Y + 1 _ V 2 + log. (V2 + 1). 

This measures the length of the logarithmic curve from the point 
B (Fig. 3), towards the left for any positive value of x; and from 
the point B towards the right, when x has any negative value from 
to -1. And, in the equation x = log y, the length of the curve is 

VT+T 2 - log Vl+ y 2 + l - V2 + log (V2 +1), 
where y is the ordinate D E, and x the abscissa A E. 

40. To determine the length of the curve for any system of log- 
arithms whose base is a and modulus M. The primary equation is 
y = log a (z + 1),= M-log e (x + 1). 



The secondary equation is, therefore, ?/ = y 1 + ^ 



2y 

y 2 M-y 2 

From this we get 



eM, u 



And the integral of this, as in Art. 39, is 

Substitute for y, its value in terms of (x + 1), and we have 

/ / — i ! s/i + ^d + ^ + i 

y-(\A+gi(i+«>' -§"«'. Y ,* -)+c. 

When x = 0, the curve is zero; hence, 

f / T 1, V 1 + M 2+1 \ 



THE INTEGRAL CALCULUS 33 

The length of the curve is, therefore, 



+ M ! + 



io ge M(y/i + i + i)]. 

And, in terms of log a this expression becomes 

nnrr-r-n \ 8 1 i VM 2 + (1 + a;) 2 + M ,„ r 

VM 2 + (1 + x) 2 - - log a ,\-— ^ ,, - VM 2 + 1 

2 VM 2 + (1 + x) 2 - M 

+ log a (VM 2 + 1 + M) = VM 2 + (1 + x) 2 

_ lo& VM»+(l+s)' + M _ v2 UMn + log (ViiPTT + M ). 



SECTION FIVE 

On the Integration of the Logarithms of Binomials, and other 
Complex Quantities. 

41. In considering the following logarithmic integrals as the 
measure of curvilinear areas, it is necessary to observe from what 
point in the axis the measure is reckoned. In Fig. 3, Art. 34, the 
line AC is the axis of x, and the curve cuts the axis at the point B. 
The form of the curve varies with the nature of the function; but, 
whatever may be the form of the function, the curve will cross the 
axis at that point where x (from origin A) has such a value as to 
make the whole function equal 1, because the logarithm of 1=0. 
Hence, these integrals measure the area (a) above the axis, when the 
value of x makes function x greater than 1 ; and they measure the 
area (c) below the axis, when the value of x makes function x less than 
1, but not negative. 

42. The integral of log (1 + x) = (1 + z)-log(l + x) — x. 
The integral of log(l — x) = — (1 - rr)-log(l - x) - x. 

When x. = 1, this last integral has the term 0-log 0; apparently 
irrational; but its value can be defined, thus: (1 - x)-log(l - x) 
= log (1 - x) (1-x) = log(l - x)° = log 1,= 0; hence, when x = 1, the 
whole value of this last integral = — 1. That is, the logarithm of 
an infinite number, multiplied by zero, = nothing. 

To find the integral of log(l + x 2 ), assume yiog(l + x 2 ) 
= £-log(l + x 2 ) + z, and differentiate, thus: 

2x 2 
log (1 + x 2 ) = log(l + x' 2 ) + j-x^ + d ' z '> 

hence, 

therefore, 

/ -log (1 + x 2 ) = x-log(l + x 2 ) +2v - 2x, 

when v is the arc of which x is the tangent. 
To find the integral of log (1 - x 2 ), assume 

yiog(l - x 2 ) = x-log(l - x 2 ) + z, 
and differentiate, thus, 

log (1 - x 2 ) = log (1 - x 2 ) - — - 2 + d-z; 

34 



THE INTEGRAL CALCULUS 35 

hence, 



/» ^x i 1 + x _ 

3 = f i 2 = log 2x; 

1— x 2 1 — a; 



therefore, 



flog (1 - x 2 ) = x -log (1 - x 2 ) + log 2x. 

1 — X 

This integral may also be had as follows: 
1- x 2 = (1 + x) • (1 - x) :. yiog(l - x 2 ) = flog (1 + x) 
+ yiog(l - x) = (1 + x)-log(l + x) - (1 - x) log(l - x) - 2x. 

In both forms of this last integral, there are quantities apparently 
irrational when x = 1. In the second form, it has been shown that 
(1 - x)-log (1 - x) = 0, when x = 1, and the whole integral is, 
therefore, 2 • log 2-2. 

1 + x ' (1 + x) 2 

In the first form log— becomes log ^ = 2 -log (1 + x) 

J- x j_ — x 

- log(l - x 2 ). Hence, the whole integral becomes, 

x-log (1 - x 2 ) + 2 log (1 + x) - log (1 - x 2 ) - 2x 

and when x = 1 this equals 2 -log (1 + x) - 2x = 2 -log 2 - 2, as 
above. 

43. By introducing constants into these functions, and pursuing 
the same methods, we get 

f log (a + x) = (a + x) log (a + x) - x - a • log a. 

flog (a - x) = - (a - x) log (a - x) - x + a -log a. 

yiog (a + x 2 ) = x-log (a + x 2 ) - 2x + 2Va-v, 

x 
when v is the arc of which — - is the tangent. 

flog (a - x 2 ) = x-log (a - x 2 ) + Va-log ^°L + x - 2x 

Va — x 

= 2\/a log(Va + x) - 2x, when x = y/a. 
flog (a 2 + x 2 ) = x-log (a 2 + x 2 ) - 2x + 2a- v, 



x 



when v is the arc of which - is the tangent. 



a 



a + x 



flog (a 2 - x 2 ) = x-log (a 2 - x 2 ) + a- log ^^ - 2x 

a — x 
= 2a -log (a + x) - 2x, when x = a. 

flog (a 2 + b 2 x 2 ) = x-log (a 2 + 6 2 x 2 ) - 2x + ^%, 

ox 
when w is the arc of which — is the tangent. 

a ° 

flog (a 2 - b 2 x 2 ) = x-log (a 2 - b 2 x 2 ) - 2x + f -log ^±-^ 

o a — ox 
= 2a -log (a + 6x) - 2x, when ox = a. 



36 THE INTEGRAL CALCULUS 

It will be observed that when x has such a value as to make the 
whole function = 0, the integral contains two logarithmic terms, which, 
when taken together, can always be valued; but not when taken 
separately, because each involves the logarithm of an infinite num- 
ber. Therefore, the whole area between the curve and its asymptote 
can be determined for each of these functions. 

The asymptote to the curve in each of these functions is the 
perpendicular dropped from that point of the axis where x has such 
a value as to make the whole function equal nothing. 

44. Theorem. Let w equal any function of x, simple or complex, 
algebraic or transcendental, then the integral, with respect to x, of 

the product of -=— and \og m w will equal the integral of \og m (w) 

with respect to w treated as a simple algebraic variable. That is, 

I -=- -\og m W = J'\og rn (w). 

The forms of this last integral, for particular values of m, may be 
had from Art. 35; for example, 

i -.i dw i 

let m = 1, and w = x + sin x. -=- = 1 + cos x: 

ax 

then 

f (1 + cos x) -log (x + sin x) = (x + sin x) Tog (x + sin x) 
— (x + sin x) + 1. 
Let 

-. o dw n 

w = 1 + x 2 . —r = 2x, 
ax 

then 

/2a; -log (1 + x 2 ) = (1 + x 2 ) -log (1 + x 2 ) - x 2 . 

Let 

dw H „ 

w = tan-z. -7- = 1 + tan 2 x: 
ax 

then 

Si\ + tanV) log tan x = tan -a; (log tan x) - tan x + 1. 

Let 

. dw 1 

W = log X. t~ - — j 

ax x 
then 

/ (--log of log a: J = log -a; (log of log a;) - log x + 1. 

Let 

m = 2, and w = 1 + x 3 , 



THE INTEGRAL CALCULUS 37 

then 

./•3a; 2 - log 2 (1 + x 3 ) = (1 + z 3 ) -log 2 (l + x 3 ) - (2 + 2x 3 ) -log (1 + z 3 ) 

-2z 3 . 

45. The integrals of certain classes of algebraic fractions may be 
readily had by means of the foregoing theorem. 

Let 

/-. \ dw . 

w = (1 + x m ). ~r = rri'X m ~ l : 
ax 

and, by the theorem, 

fm- x m ~ l -log (1 + x m ) = (1 + x m ) -log (1 + x m ) - x m . 

Assume this equals z w -log (1 + x m ) + z, and differentiate. 

m-a^-log (1 + x m ) = m-a^-log (1 + x m ) + w + d-z. 

Hence, 

/» /v.2w— 1 

m 



Therefore 



m = 



//v.2w— 1 
_ _= _*,«a--log(l+a!«). 

r&^_ = lL m _ log (! + ^)Y 

And making w = (1 - # m ), by repeating the process, we have 

/~2m-l 1 / \ 
= - — [x m + log (1- x m ) 
1 - x m m\ & v V 

Giving particular values to m, we get, 

L frti = x ~ log (1 + x) - 

f^ = - (x + log (1 - *)) 

/?=* "i + log (* " i) 

|- /Y^-r = f-^-p = 2f Vx - log (1 + Vx)} 

f — l —^ = - 2 (Vx + log (1 - Vx)). 
J 1 - Vx 



m = — 



m = 2 



38 THE INTEGRAL CALCULUS 

/^ = 2 (^ +log(1 -^> 

/ £2^5-1 1 / _ \ 

rr^ = vs r " og (1 + * v v 

46. The general formulae given in the preceding article may be 
varied as follows: 

r x m ~ l l , ,„ N 

I q = —-log (1 + x m ). 

J 1 + x m m & v y 

r x m - 1 1 1 /-, m n 

I = log (1 - x m ). 

J 1 - x m m & v ' 

/— = — X 2m [X m - log (1 + X m ) J 
1 + x OT 2m m\ & v '/ 

/z 3 ™ -1 11/ \ 
= -— x 2m [x m + log (1 - x m ) 
1 - x m 2m m V / 

/ x im-l i / \ i i 
= — z m - log (1 + X m ) ) -7T- X 2m + zz- X? m . 
1 + x m m V / 2m 3m 

/z 47 " -1 1 / \\ 1 1 
= [x m + log (1 - x m ) - jr— z 2m - — z 3rn . 
1 - z w ?ti V V 2m 3m 

47. Let 

w = a + foe™ + cz n • -7— = mbx m ~ l + 7icz n_1 . 

By the theorem, 

f(mbx m - 1 + Tier" -1 ) -log (a + bx m + cz") 

= (a + bx m + ex") -log (a + bx m + ex") - (a + bx m + ex") + 1. 

Assume this integral equals 

(bx m + cx n ) - log (a + bx m + ex") + z 

and differentiate. 
Then 

*mb 2 x 2m ~ l + (m + n) 6cx m+n_1 + nc 2 x 2n ~ l 



r 



a + bx m + ex 71 
= 6z m + cx n - a-log(a + bx m + cx n ) + a -log a. 



= -2 



THE INTEGRAL CALCULUS 39 

The constants a, b, c, m, n may be positive or negative, whole num- 
bers or fractions. 

To give an example of this last integral: 

Let 

a = 1, b = -2, c = 3, m = -4, n = 5; 

then 



j 



( 40X 9 O ~ _ 



The foregoing methods may be applied to any function of two or 
more variables, however complex it may be, provided the function 
contains at least one constant not affected by the variable. 

48. The theorem under consideration may also be used to find the 
integrals of certain classes of fractions containing circular functions. 

Let v represent any circular arc in the first quadrant, and s = sin v, 
and t = tan v. 

Take the function w = (a + b sin m y), 



av 



Then, by the theorem, 



/Vl - s 2 -bm-s m - 1 -log (a + bs m ) = (a + bs m ) log (a + bs m ) 
- (a + bs m ) + 1. 

Assume this equals 6s m -log (a + bs m ) + g, and differentiate; then 



v/1 — S 2 'b 2 S 2m ~ l 

= - z = bs m - a • log (a + bs m ) + a • log a. 



J a + 6s 

Take the function w = a + b tan m v. 

^- = &m(* m - 1 + * m+1 ). 

Then, by the theorem, 

fbm(t m - 1 + t m+1 ) • log(a + bt m ) = (a + ^ m ) ■ log(a + &r) - (a + &P) + 1. 

Assume this equals bt m • log(a + bt m ) + z, and differentiate. 

Then 

b 2 -m(t 2m ~ l + t 2m+1 ) 



f 



a + fa* 



= -z = bt m - a-log(a + bt m ) + a log a. 



49. To find the integral of (1 + 2x) -log (1 - z 3 ), assume it equals 
(x + a; 2 ) log (1 - x s ) + z and differentiate; then 

'3z 3 + 3z 4 



/ "3s 3 + 



2 = 



40 THE INTEGRAL CALCULUS 

By the common rules, this quantity is not integrible in its present 

form; but, add to it - = and we have : ^ , and this 

1 - x 3 1 - x 3 

3x 2 
equals - Here, both the sum and the added quantity are 

separately integrible, thus: 
3z 2 



1 — x 



= - log (1 - X s ). 



/ _ 

fj=i - -( 3x + |^ 2 + 3 log (1 - *)) 

Therefore, 

z = log (1 - a; 3 ) - 3x - fz 2 - 3 log (1 - x); 
and 
f(l + 2x) -log (1 - z 3 ) = (1 + x + a: 2 ) log (1 - a; 3 ) - 3 log (1 - x) 

Q/v. 3/V.2 

— tJJU — "2"«* / • 

To measure the whole area between the curve and its asymptote, 
that is, to ascertain the value of this integral when x = 1, it is neces- 
sary to determine the value of the two irrational terms, 3 log (1 — x 3 ) 
and -3 log (1 - x), when x = 1. 

In Art. 42, this was effected by making the quantity within the 

vinculum the same in each logarithmic term. This cannot be done 

in the present case; but the value may be obtained by means of the 

ultimate ratio; thus, 

Let 

x = 1 - a, 

then, 

1 — x = a 

1 - x 3 = 3a - 3a 2 + a 3 , 

when x approaches to 1, a becomes infinitely small; so the ultimate 
ratio 1 - x to 1 - z 3 is 1 to 3, and the difference of the logarithms 
of these two quantities is, therefore, log 3. 

That is, log (1 - x 3 ) - log (1 '— x) = + log 3, when x = 1. Hence, 
the whole area between the curve and its asymptote is 3 log (1 - a: 3 ) 

- 3 log (1 - x) - 3x - fr* = 3 log 3-4-5. In like manner, the 

/3#3 _ 3^.4 
1 3 • 

And 

Sx 3 - 3Z 4 3z 5 3z 3 

1 +x 3 + 1 +z 3 ~ 1 +x' 

Fere the last two terms are separately integrible, therefore, 

f{\ - 2x) log (1+ x 3 ) = (x - 1 - x 2 ) log (1 + x 3 ) + 3 log (1 + x) 

— Zx + %x 2 . 



THE INTEGRAL CALCULUS 41 

And, when m is any positive whole number greater than 1, the 
integral of 

- (1 + 2x + 3x 2 + • • • (m - l)x m - 2 ) -log (1 - x m ) = (1 + x + x 2 + r 3 
_l ajm-i) 4 0g (i _ a-m) - m .\ g (i - x ) - mix + Jz 2 + \x z 

1 

+ -a;" 1 - 1 ); 

m - 1 

and the whole area between the curve and its asymptote, that is, 
the value of this integral when x = 1, is 

ra-(logra - 1 - - - -}. 

\ 2 3 m - 1/ 



